1. Fractals in nature

2. A fractal fern

3. A fractal tree

4. How to grow a digital tree?

5. A fractal is an object with a fractional dimension!

7. 0.6039

8. Other example of fractal: Koch’s snowflake
D=log4/log3=1.261

9. Self-similarity in Koch’s curve

10. Two “classic” examples of fractal:
the Julia set and
the Mandelbrot set

11. How to create a Julia set?
Consider the map
f: z --> z^2 + c
where z = x + iy = (x, y) and c = a + ib = (a, b) is a parameter in the mapping.
It is equivalent to the two-dimensional map
(Polar coordinate)
r eiθ--> r^2 e2iθ+ c

12. This map of the complex numbers is equivalent to 3 successive transformations on the complex plane.
Stretch points inside the unit circle towards the origin. Stretch points outside towards infinity
Cut along the positive x-axis. Wrap the plane around itself once by doubling every angle.
Shift the plane over so the origin lies on (a, b).

13. For other values of c we need a computer to find out the fixed points
Despite all this stretching, twisting, and shifting there is always a set of points that transforms into itself.
Such sets are called the Julia sets (after the French mathematician Gaston Julia who discovered them in the 1910s.)
The Julia set for c = (0, 0) is easy to find: the set is the unit circle.
For other values of c we need a computer to find out the fixed points

14. Examples of
the Julia set
on z plane

15. A Julia set is either
totally connected or
totally disconnected!

16. Self-similarity of the
Julia set

17. An artistic visualization of the Julia set

18. Whether a Julia set is connected or not depends on the parameter c.
Plot the Julia sets for all parameter values c.
If the value of c makes the Julia set connected, then we say this c belongs to
the Mandelbrot set. We can plot the Mandelbrot set on the c plane.
(Note: the Julia set is defined on the z plane)
Examine the Julia set to determine whether it is connected or not takes a long time. Luckily, we need to study only one point in the z plane: the origin
If the origin never escapes to infinity then it is either a part of the Julia set or is trapped inside it. In both cases, the Julia set is connected. (Mandelbrot)
(Note: If the origin is part of the set, the set is dendritic (branch-like). If it is trapped inside the set, the set is topologically equivalent to a circle.)

19. Mandelbrot set on the c plane
(x,y)=(-2,0)
(x,y)=(1/4,0)
(x,y)=(-3/4,0)
(x,y)=(0,0)

20. the bifurcation diagram!
Mandelbrot set and
the bifurcation diagram! 3 4 5 4 2 1 3 8

21. The first computer print-out of the Mandelbrot set
All the ”islands” in the set are connected!!

22. The fascinating “universe” of the Mandelbrot set

36. The end

37. “Bulbs” with different periods

38. Period 3 3

39. Period 4 4

40. Period 5 5

41. Period 7 7

42. You can find thousands of artistic fractals on the web, for example...

44. etc...